
where \(a_0\)  and \(a_1\)  denote respectively the value of the scale factor today and at the onset of oscillations, \(a_1\equiv a(T_1)\) .Whenever the initial misalingment angle is large, the relic density in Eq: relic density simple ALP a needs to be corrected by the so-called anharmonicity factor, see sec:anharmonicityfunction.
The latter can be expressed in turn in terms of the axion mass using Friedmann equations and the conservation of entropy, \(a_1\propto 1/\sqrt{m_a M_{\rm Pl}}\) , allowing us to obtain the ratio of the axionic relic density to the current DM abundance \(\rho _{\rm DM}\)  as a function of \(m_a\) , \(\theta _1\)  and \(f_a\) ,The slight discrepancy with respect to the result in Ref. [1]} is due to the use of the updated value of \(\rho _{\rm DM}\simeq 1.26\, {\mathrm {keV}}/{\mathrm {cm}^{3}}\)  from Planck 2018 data [2]} and a different choice for the onset of the oscillations that better fits the numerical result
(\(1.67 \, H=m_a\)  instead of \(3H=m_a\) , see e.g. Refs. [3]}, [4]}).
\(\frac{\rho _{a,0}}{\rho _{\rm DM}} \simeq 5.7 \sqrt{\frac{m_{a}}{\mathrm {eV}}} \left(\frac{\theta _{1}\,f_a}{ 10^{12}\, \mathrm {GeV}}\right)^{2} \mathcal {F}(T_{1})\,,\) 
